The Constrction Of The Generalized Birkhoff Equations Of Nonholonomic Systems And Its Noether Theory

Birkhoff system dynamics is a very important research direction in analytical mechanics.Any kind of local,regular,analytic,holonomic dynamical systems,has a first-order self-adjoint Birkhoffian formulation,and then has a relatively complete integral theory.But the study of Generalized Birkhoff system,is not perfect.Nonholonomic mechanical systems in the generalized Birkhoff system is widely used in various types of vehicles,train cars,aircraft landing gear,mobile robots,motion constrained robot wheel systems,and electrical machinery,hydraulic machinery and other engineering mechanical systems.The study of nonholonomic mechanics system is important value to theory and practice.The characterization of generalized Birkhoff nonholonomic mechanical systems is a very effective way to provide an idea and methods to solve the practical problems of nonholonomic mechanical systems.Noether symmetry and Lie symmetry theory are very important ways for solving Birkhoff Equations.They also have wide application in the generalized Birkhoff equations.Based on Birkhoff system dynamics,the generalized Birkhoffian realization of nonholonomic mechanical systems is researched,and generalized Birkhoff equations without additional item are constructed,and the derivation of the Noether symmetry theory for such generalized Birkhoff system is completed.Those make a point of exploration and supplementary for the basic framework of the generalized Birkhoff system.The main work is as follows:1.With a large number of domestic and foreign references for the foundation,the Lagrange equationn of the Chaplygin system and a general order nonholonomic system is reduced to the first-order self-adjoint Birkhoffian formulation.We obtain the generalized Birkhoff equations of nonholonomic systems.It is the same as the form of Birkhoff equation.Its dimension depends on the number of independent variables and the integrity constraint,nonholonomic constraint: when the nonholonomic constraint number is an odd number,the equation does not meet the conditions of closure and regularity.2.Based on the Birkhoff system dynamics research,put forward the suitable method,for constructing the generalized Birkhoff equation of nonholonomic systems,and point out the difficulties of various methods and its advantages and disadvantages.Taking Appell-Hamel as an example,construct its generalized Birkhoff equation.3.The generalized Birkhoff Noether system theory is deduced,which is similar to the Noether theorem of Birkhoff system.The theory includes obtaining the first integrals from the known symmetry,quasi-symmetry and generalized quasi symmetry,and then give a simple proof of it.4.Noether’s inverse problems of such generalized Birkhoff system are studied.Based on solving the Birkhoff equation of its transformation,a group of simple equation,which contains the group generator function,is obtained.The group generator function corresponds symmetry,quasi-symmetry and generalized quasi symmetry.